The 1-2-3 conjecture for uniform hypergraphs

Let $r\geq 3$ and let $\mathcal{H}$ be an $r$-uniform hypergraph, meaning that every edge has size $r$. A hypergraph vertex coloring is proper when every edge contains at least two vertices with distinct colors. For an edge weighting $w:E(\mathcal{H})\rightarrow\{1,2,\ldots,k\}$, let $\chi^{e}(\mathcal{H})$ be the least $k$ for which the induced vertex coloring is proper. The 1-2-3 conjecture for uniform hypergraphs. If $\mathcal{H}$ has no isolated edge, then $\chi^{e}(\mathcal{H})\leq 3$. This extends the 3-uniform hypergraph conjecture to every uniformity $r\geq 3$. The source gives the bound $\chi^{e}(\mathcal{H})\leq\max\{5,r+1\}$ in general, but leaves the conjectured bound 3 unresolved.